Errors in Jacobian Elliptic Functions
- To: mathgroup at smc.vnet.net
- Subject: [mg7629] Errors in Jacobian Elliptic Functions
- From: seanross at worldnet.att.net
- Date: Fri, 20 Jun 1997 16:16:02 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
I have an application which requires extensive use of Jacobian Elliptic Functions like JacobiSN, EllipticPi and EllipticF. I have found that these functions have some very small errors near the ends of their ranges. For example, JacobiSN is supposed to be bounded by +1 and -1, just like Sine and Cosine. However, for JacobiSN[x,p], with p very close to zero, JacobiSN can occasionally come out slightly greater than +1. The problem is that my application also requires that I take ArcSin of JacobiSN so that numbers slightly greater than +1 return complex values which mess everything up. I have done some study on these errors and they are not continuous functions of the arguments. I can plot regions as wide as 10^-5 where they have the error and then find continuous regions with no errors. Question #1. Does anyone know if it is better to truncate the JacobiSN's with something like Min[Max[JacobiSN[x,p],-1],1] or to rescale the whole thing with something like JacobiSN[x,p]/(1+10^-11)? The second manifestation of this error is in the EllipticPi and EllipticF functions when the first argument is very large or the second argument is very close to Pi/2. I get obvious wrong answers for certain arguments in this region and extraneous complex values in others. Question #2. Does anyone know of a way to estimate the error in an ill-behaved function like this? Thanks, Sean Ross